DISCUSSION FORUM On Rogers Hollingsworth and Karl H. Mu¨ller—‘Transforming Socio-Economics with

Socio-Economic Review Advance Access published August 13, 2008
Socio-Economic Review (2008) 1–38
On Rogers Hollingsworth and Karl H.
Müller—‘Transforming Socio-Economics with
a New Epistemology’, Socio-Economic
Review, 6 (2008), 395 – 426
Keywords: socio-economics, epistemology, evolution, complex networks,
self-organization, power law, interdisciplinarity, multi-level analysis
JEL classification: D8 information, knowledge, uncertainty, D85 network
formation and analysis, theory
The quest for theoretical foundations
of socio-economics: epistemology,
methodology or ontology?
Robert Boyer
PSE – Paris-School of Economics, EHESS, CEPREMAP, Paris, France
Correspondence: [email protected]
The article under review has two sources of inspiration, one negative and one
positive. On the negative side, the authors express their dissatisfaction with the
current state of the socio-economic community with an implicit reference to
the Society for the Advancement of Socio-Economics (SASE). Socio-economic
research is characterized by strong empirical and comparative studies as well as
a relentless criticism of the neoclassical theory but no relevant theorizing
(395 – 396). On the positive side, Hollingsworth and Müller perceive a window
# The Author 2008. Published by Oxford University Press and the Society for the Advancement of Socio-Economics.
All rights reserved. For Permissions, please email: [email protected]
Discussion forum
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Gibbons, M., Limoges, C., Nowotny, H., Schwarztman, S., Scott, P. and Trow, M. (1994)
The New Production of Knowledge: The Dynamcis of Science and Research in
Contemporary Societies, London, Sage.
Nowotny, H., Scott, P. and Gibbons, M. (2001) Re-Thinking Science. Knowledge and the
Public in an Age of Uncertainty, Cambridge, UK, Polity Press.
Nowotny, H. and Testa, G. (in press) Die gläsernen Gene. Gesellschaftliche Optionen im
molekularen Zeitalter, Frankfurt a.M., Suhrkamp Verlag.
Pickstone, J. (2007) ‘Working Knowledges Before and After Circa 1800. Practices and
Disciplines in the History of Science, Technology, and Medicine’, Isis, 98, 489–516.
Science (2008) ‘Special Section: Cities’, 319, 739 –775.
Interdisciplinarity in socio-economics,
mathematical analysis and the predictability
of complex systems
Didier Sornette
ETH Zurich, Department of Management, Technology and Economics, Zurich, Switzerland
Correspondence: [email protected]
The essay by Hollingworth and Müller (2008) on a new scientific framework
(Science II) and on the key role of transfers across disciplines makes fascinating
reading. As an active practitioner of several scientific fields (earthquake physics
and geophysics, statistical physics, financial economics, and some incursions
into biology and medicine), I witness every day first-hand the power obtained
by the back-and-forth transfer of concepts, methods and models occurring in
interdisciplinary work and thus applaud the formalization and synthesis
offered by Hollingworth and Müller (2008). Section 2 presents a personal highlight of my own scientific path, which illustrates the power of interdisciplinarity
as well as the unity of the mathematical description of natural and social
But my goal is not just to flatter Rogers and Karl and praise their efforts. I wish
here to suggest corrections and complements to the broad picture painted in
Hollingworth and Müller (2008). I will discuss two major claims in some
detail. First, in Section 3.1, I take issue with the claim that complex systems are
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Transforming Socio-Economics with a New Epistemology
in general ‘not susceptible to mathematical analysis, but must be understood by
letting them evolve—over time or with simulation analysis’. In Section 3.2,
I present evidence of the limits of the claim that scientists working within
Science II do not make predictions about the future because it is too complex.
I conclude with Section 4, in which I point out a possible missing link between
Science I and Science II, namely ‘Quantum Science’, and the associated conceptual and philosophical revolution. I also tone down the optimism echoed by
Hollingworth and Müller (2008) that the approaches in terms of complex
networks will allow for a stronger transfer of theoretical models across widely
disparate fields; in particular, between the natural and social sciences.
A personal highlight illustrating the power of
interdisciplinarity and unity of the mathematical description of
natural and social processes
Let me illustrate with a personal anecdote how the power of interdisciplinarity
can go beyond analogies to create genuinely new paths to discovery. In the
example I wish to relate, the same fundamental concepts have been found to
apply efficiently to model, on the one hand, the triggering processes between
earthquakes leading to their complex space– time statistical organization
(Helmstetter et al., 2003; Ouillon and Sornette, 2005; Sornette and Ouillon,
2005) and, on the other hand, the social response to shocks in such examples
as Internet downloads in response to information shocks (Johansen and Sornette,
2000), the dynamic of sales of book blockbusters (Sornette et al., 2004; Deschatres
and Sornette, 2005) and of viewers’ activity on YouTube.com (Crane and Sornette, 2007), the time response to social shocks (Roehner et al., 2004), financial
volatility shocks (Sornette et al., 2003) and financial bubbles and their crashes
(Johansen et al., 1999, 2000; Sornette, 2003; Andersen and Sornette, 2005;
Sornette and Zhou, 2006). The research process developed as follows.
First, the possibility that precursory seismic activity, known as foreshocks,
could be intimately related to aftershocks has been entertained by several
authors in the past decades, but has not been clearly demonstrated by a combined
derivation of the so-called direct Omori law for aftershocks and of the inverse
Omori for foreshocks within a consistent model. In a nutshell, the Omori law
for aftershocks describes the decay rate of seismicity after a large earthquake
(called a mainshock), roughly going as the inverse of time since the mainshock.
The inverse Omori law for foreshocks describes the statistically increasing rate of
earthquakes going roughly as the inverse of the time till the next mainshock. The
inverse Omori law has been demonstrated empirically only by stacking many
earthquake sequences (see Helmstetter and Sornette, 2003 and references
therein). We had been working for several years on the theoretical understanding
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of a statistical seismicity model, known at the ETAS model, a self-excited Hawkes
conditional point process in mathematical parlance. This model or its siblings are
now used as the standard benchmarks in statistical seismology and for evaluating
other earthquake forecast models (Jordan, 2006; Schorlemmer et al., 2007a, b).
We had the intuition that the inverse Omori law for foreshocks could be
derived from the direct Omori law by viewing mainshocks as the ‘aftershocks
of foreshocks, conditional on the magnitude of mainshocks being larger than
that of their progenitors’. But we could not find the mathematical trick to
complete the theoretical derivation. In parallel, we were working on the statistical
properties of financial returns and were starting a collaboration with J.-F. Muzy,
one of the discoverers of a new stochastic random walk with exact multifractal
properties, named the multifractal random walk (MRW; Bacry et al., 2001;
Muzy and Bacry, 2002), which seems to be a powerful model of financial time
series. We then realized that similar questions could be asked on the precursory
as well as posterior behaviour of financial volatility around shocks. The analysis of
the data showed clear Omori-like and inverse Omori-like behaviour around both
exogenous (11 September 2001, or the coup against Gorbachev in 1991) and
endogenous shocks. It turned out that we were able to formulate the solution
mathematically within the formalism of the MRW and we showed the deep
link between the precursory increase and posterior behaviour around financial
shocks (Sornette et al., 2003). In particular, we showed a clear quantitative
relationship between the relaxation after an exogenously caused shock and the
relaxation following a shock arising spontaneously (termed ‘endogenous’).
Then, inspired by the conceptual path used to solve the problem in the financial
context, we were able to derive the solution in the context of the ETAS model,
demonstrating mathematically the deep link between the inverse Omori law
for foreshocks and the direct Omori law for aftershocks in the context of the
ETAS model (Helmstetter et al., 2003; Helmstetter and Sornette, 2003). The
path was simpler and clearer for financial time series, and their study clarified
the methodology to be used for the more complicated specific point processed
modelling earthquakes.
This remarkable back-and-forth thought process between the two a priori very
different fields will remain a personal highlight of my scientific life.
On self-organizing processes and multi-level analysis
The emphasis of Hollingworth and Müller (2008) on self-organizing processes
and multi-level analysis to comprehend the nature of complex social systems is
welcomed as it indeed reflects an important strategy used by researchers. But
more problematic is the endorsement of the claims, which are variations of
a common theme, that
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Transforming Socio-Economics with a New Epistemology
(1) ‘increasingly analysts maintain that such systems are not susceptible to mathematical analysis, but must be understood by letting
them evolve—over time or with simulation analysis’ (p. 399),
(2) ‘the emerging perspective, rapidly diffusing across academic
disciplines, suggests that the world does not change in predictable
way’ (p. 398),
(3) ‘hardly any scientist in these fields is able to make successful predictions about the future, as self-organizing processes are understood
best by retrospective analysis’ (p. 403).
Hollingworth and Müller (2008) give thus resonance to a view upheld by various
groups in different communities, which I find misguided and dangerous, while
unfortunately widespread.
On models of complex systems
Let me first address claims (1) and (2), perhaps best personified by Stephan
Wolfram and elaborated in his massive book entitled ‘A New Kind of Science’
(Wolfram, 2002). According to Wolfram, the most interesting problems presented
by nature (biological, physical and societal) are likely to be formally undecidable
or computationally irreducible, rendering proofs and predictions impossible. Take
the example of the Earth’s crust and the problem of earthquake prediction or the
economies and financial markets of countries and the question of predicting their
recessions and their financial crashes. Because these events depend on the delicate
interactions of millions of parts, and seemingly insignificant accidents can sometimes have massive repercussions, it is argued that their inherent complexity
makes such events utterly unknowable and unpredictable. To understand precisely
what this means, let us refer to the mathematics of algorithmic complexity (Chaitin,
1987), which provides one of the formal approaches to the study of complex
systems. Following a logical construction related to that underpinning Gödel’s
(1931) incompleteness theorem, most complex systems have been proved to be
computationally irreducible, i.e. the only way to decide about their evolution is
to actually let them evolve in time. The only way to find out what will happen is
to actually let it happen. Accordingly, the future time evolution of most complex
systems appears inherently unpredictable. Such statement plays a very important
role in every discussion on how to define and measure complexity.
However, it turns out that this and other related theorems (see Chaitin, 1987
and Matthew Cook in Wolfram, 2002) are useless for most practical purposes and
are in fact misleading for the development of scientific understanding. And the
following explains why. In a now famous essay entitled ‘More Is Different’, Phil
Anderson (1972), 1977 Nobel Prize winner and a founder of the Santa Fe Institute
of complexity, described how features of organization arise as an ‘emergent’
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property of systems, with completely new laws describing different levels of magnification. As a consequence, Physics, for instance, works and is not hampered
by computational irreducibility. This is because physicists only ask for answers at
some coarse-grained level (see Buchanan, 2005 for a pedagogical presentation of
these ideas). In basically all sciences, one aims at predicting coarse-grained properties. Only by ignoring most molecular detail, for example, did researchers
ever develop the laws of thermodynamics, fluid dynamics and chemistry, providing
remarkable tools for explaining and predicting new phenomena. From this perspective, one could say that the fundamental theorems of algorithm complexity
are like pious acts of homage to our intellectual ancestors: they are solemnly
taken out, exhibited and solemnly put away as useless for most practical applications. The reason for the lack of practical value is the focus on too many
details, forgetting that systems become coherent at some level of description. In
the same vein, the butterfly effect, famously introduced by Lorenz (1963, 1972)
to communicate the concept of sensitivity upon initial conditions in chaos, is actually not relevant in explaining and predicting the coherent meteorological structures at large scales. As a result of the spontaneous organization of coherent
structures (Holmes et al., 1998), there is actually predictability in meteorology
and climate as well as in many other systems, at time scales of months to years,
in apparent contradiction with the superficial insight provided by the butterfly
These points were made clear by Israeli and Goldenfeld (2004, 2006) in their
study of cellular automata, the very mathematical models that led Wolfram to
make his grand claims that science should stop trying to make predictions and
scientists should only run cellular automata on their computers to reproduce,
but not explain, the complexity of the world. Cellular automata are systems
defined in discrete Manhattan-like meshed spaces and evolve in discrete time
steps, with discrete-valued variables interacting according to simple rules.
These remarkably simple systems have been shown to be able to reproduce
many of the behaviours of complex systems. In particular, it is known that
most of them are ‘universal Turing computational machines’, i.e. they are
capable of emulating any physical machine. Because they can emulate any
other computing device, they are therefore undecidable and unpredictable. But
which of the systems are capable of universal computation is not generally
known. In this respect, one results stands out for our purpose. Matthew Cook,
whose theorem is reported in Wolfram (2002), showed that one simple cellular
automaton, known as ‘rule 110’ in Wolfram’s nomenclature of one-dimensional
cellular automata with nearest-neighbour interaction rules, is such a universal
Turing machine.
Now, Israeli and Goldenfeld applied a technique called ‘renormalization
group’ (Wilson, 1999) to search for what could be the new laws, if any, that
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Transforming Socio-Economics with a New Epistemology
describe the coarse-grained average evolution of such cellular automata. Technically, the new laws are determined by a self-consistency condition that (i) coarsegraining the initial conditions and applying the new laws should provide the same
final description and (ii) letting the system evolve according to the true microscopic laws and then coarse-graining the resulting pattern. By coarse-graining,
one focuses only on the most relevant details of the pattern-forming processes.
Israeli and Goldenfeld established that computationally irreducible cellular automata become predictable and even computationally reducible at a coarse-grained
level of description. The resulting coarse-grained cellular automata that they constructed by coarse-graining different cellular automata were found to emulate the
large-scale behaviour of the original systems without accounting for small-scale
details. In particular, rule 110 was found to become a much simpler predictable
system upon coarse-graining. By developing exact coarse-grained procedures on
computationally irreducible cellular automata, Israeli and Goldenfeld have
demonstrated that a scientific predictive theory may simply depend on finding
the right level for describing the system. For physicists, this is not a surprise:
by asking only for approximate answers, Physics is not hampered by computational irreducibility, and I believe that this statement holds for all natural and
social sciences with empirical foundations.
On predictability of the future in complex systems
Let me now turn to the third claim cited in the above introduction of Section 3
that ‘hardly any scientist in these fields is able to make successful predictions
about the future’, and more generally that predicting the future from the past
is inherently impossible in most complex systems. This view has recently been
defended persuasively in concrete prediction applications, such as in the socially
important issue of earthquake prediction (see e.g. the contributions in Nature
debate [1999]). In addition to the persistent failure in reaching a reliable earthquake predictive scheme up to the present day, this view is rooted theoretically
in the analogy between earthquakes and self-organized criticality (Bak, 1996).
Within this ‘fractal’ framework, there is no characteristic scale and the power-law
distribution of earthquake sizes suggests that the large earthquakes are nothing
but small earthquakes that did not stop. Large earthquakes are thus unpredictable
because their nucleation appears not to be different from that of the multitude of
small earthquakes.
Does this really hold for all features of complex systems? Take our personal
lives. We are not really interested in knowing in advance at what time we will
go to a given store or drive in a highway. We are much more interested in forecasting the major bifurcations ahead of us, involving the few important things,
like health, love and work, that count for our happiness. Similarly, predicting
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the detailed evolution of complex systems has no real value, and the fact that we
are taught that it is out of reach from a fundamental point of view does not
exclude the more interesting possibility of predicting the phases of evolutions
of complex systems that really count (Sornette, 1999).
It turns out that most complex systems around us do exhibit rare and sudden
transitions which occur over time intervals that are short compared with the
characteristic time scales of their prior or posterior evolution. Such extreme
events express more than anything else the underlying ‘forces’ usually hidden
by almost perfect balance and thus provide the potential for a better scientific
understanding of complex systems. By focusing on these characteristic events,
and in the spirit of the coarse-graining metaphor of the cellular automata discussed in Section 3.1, a small but growing number of scientists are re-considering
the claims of unpredictability. After the wave of complete pessimism on earthquake prediction in the West of the 1980s and 1990s, international earthquake
prediction experiments such as the recently formed Collaboratory for the
Study of Earthquake Predictability (Jordan, 2006) and the Working Group on
Regional Earthquake Likelihood Models (Schorlemmer et al., 2007a, b) aim to
investigate scientific hypotheses about seismicity in a systematic, rigorous and
truly prospective manner by evaluating the forecasts of models against observed
earthquake parameters (time, location, magnitude, focal mechanism, etc.) that
are taken from earthquake catalogues.
Recent developments suggest that non-traditional approaches, based on the
concepts and methods of statistical and non-linear physics, could provide
a middle way to direct the numerical resolution of more realistic models and
the identification of relevant signatures of impending catastrophes, and in particular of social crises. Enriching the concept of self-organizing criticality, the predictability of crises would then rely on the fact that they are fundamentally
outliers (Johansen and Sornette, 2001), e.g. financial crashes are not scaled-up
versions of small losses, but the result of specific collective amplifying mechanisms (see Chapter 3 in Sornette, 2003, where this concept is documented empirically and discussed in the context of coherent structures in hydrodynamic
turbulence and of financial market crashes). To address this challenge, the available theoretical tools comprise in particular bifurcation and catastrophe theories,
dynamical critical phenomena and the renormalization group, non-linear dynamical systems and the theory of partially (spontaneously or not) broken symmetries. Some encouraging results have been gathered on concrete problems
[see the reviews by Sornette (2005, 2008) and references therein], such as the prediction of the failure of complex engineering structures (a challenge generally
thought unreachable by most material scientists), the detection of precursors
to stock market crashes with real advance published predictions (another unattainable challenge generally according to most financial economists) and the
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Transforming Socio-Economics with a New Epistemology
prediction of human parturition and epileptic seizures, to cite some subjects I
have been involved with, with exciting potential for a variety of other fields.
Other pioneers in different disciplines are slowly coming to grip with the
potential for a degree of predictability of extreme events in many complex
systems (see, for instance, the chapters in Albeverio et al., 2005). Let us also
mention Jim Crutchfield who proposes that connections between the past and
future could be predicted for virtually any system with a ‘computational mechanics’ approach based on sorting various histories of a system into classes, so
that the same outcome applies for all histories in each class (Ay and Crutchfield,
2005; Crutchfield and Görnerup, 2006). Again, many details of the underlying
system may be inconsequential, so that an approximate description much like
Isreali and Goldenfeld’s coarse-grained cellular automata models can be organized and used to make predictions.
Agent-based models developed to mimic financial markets have been found to
exhibit a special kind of predictability. While being unpredictable most of the
time, these systems show transient dynamical pockets of predictability in which
agents collectively take predetermined courses of action, decoupled from past
history. Using the so-called minority and majority games as well as real financial
time series, a surprisingly large frequency of these pockets of predictability have
been found, implying a collective organization of agents and of their strategies
which ‘condense’ into transitional herding regimes (Lamper et al., 2002;
Andersen and Sornette, 2005). Again, grand claims of intrinsic lack of predictability seem to me like throwing out the baby with the bath water, forgetting that the
heterogeneous nature in space and time of the self-organization of complex
systems does not exclude partial predictability at some coarse-grained level.
Let me end this discussion by extrapolating and forecasting that a larger multidisciplinary integration of the physical and social sciences together with artificial
intelligence and soft-computational techniques, fed by analogies and fertilization
across the natural and social sciences, will provide a better understanding of the
limits of predictability of crises.
Concluding remarks on quantum decision theory and the
theory of networks
I would like to conclude with two remarks.
Hollingworth and Müller (2008) contrast the ‘old’ Descartes – Newton Science
I with the new Science II framework which emphasizes concepts such as complex
adaptive systems, self-organization and multi-scale patterns, scale invariance,
networks and other buzzwords. I was surprised not to see discussed another
Science the ‘Quantum Science’ emerging from the scientific and philosophical
revolution triggered by the understanding that Nature works through the
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agency of fundamentally quantum mechanical laws, which have very little to do
with the macroscopic laws apparent directly to our five perception senses. In my
view, for a variety of disciplines, but perhaps not yet for the social sciences,
quantum mechanics has had more impact than Science II. At the ontological
level, Quantum Science has had a tremendous influence in all fields, by providing
a fundamental probabilistic framework, rooted in the Heisenberg uncertainty
principle, the intrinsic non-separability theorem and the existence of intrinsic
sources of noise and energy in the fluctuations of the ‘void’ (showing that the
void does not exist ontologically). This has attacked, much more deeply than,
for example, the theory of chaos ever has, the misconception that future scenarios
are deterministic and fully predictable.
In this concluding section, I would like to suggest that Quantum Science may
enjoy a growing impact in the social sciences, via the channel of decision making
operating in humans by emphasizing the importance of taking into account the
superposition of composite prospects, whose aggregated behaviour form the
structures, such as society and economies, that scholars strive to understand.
In a preliminary essay, Slava Yukalov and I have introduced a ‘quantum decision
theory’ of decision making based on the mathematical theory of separable
Hilbert spaces on the continuous field of complex numbers (Yukalov and
Sornette, submitted), the same mathematical structure on which quantum mechanics is based. This mathematical formulation captures the effect of
superposition of composite prospects, including many incorporated intentions,
which allows us to describe a variety of interesting fallacies and anomalies
that have been reported to characterize the decision-making processes of real
human beings.
My second remark concerns the claim that ‘complex networks allow for
a transfer of theoretical models across widely disparate fields’. I am afraid that
the optimism that the theory of complex networks will play such a special role
is nothing but more hype, somewhat in the lineage of those in the last decades
that involved buzzwords such as fractals, chaos, and self-organized criticality.
They all had their period of fame and excesses, followed by maturation
towards a more reasonable balanced position within the grand edifice of
science. With Max Werner, we have recently commented on the limits of applying
network theory in the field of earthquake modelling and predictability (Sornette
and Werner, 2008) and I believe much the same criticisms would apply to the use
of network theory in the social sciences. With Malevergne and Saichev, we have
developed this in theoretical synthesis (Malevergne et al., submitted) showing in
particular that the mechanism of ‘preferential attachment’, at the basis of the
understanding of scale-free networks found in social networks, the world-wide
web or networks of proteins reacting with each other in the cell, is nothing but
a rediscovery and rephrasing in a slightly different language of the famous
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model of incoming and growing firms developed by Simon in 1955, based on the
Gibrat principle of proportional growth (Gibrat, 1931). The ‘new’ science of networks thus has deep roots in economics! Viewing the rather unsophisticated level
of many discussions on power laws and other statistical regularities reported in
the ‘new’ science of networks, while not disputing the existence of significant progress in network theory, I wonder whether this ‘new’ science would not profit
from a better reading of the best works in economics of the twentieth century.
Ending on a more positive note, this illustrates my fervent faith in the power
of interdisciplinarity, practiced with the rigor and diligence necessary to ensure
depth and fecundity.
Albeverio, S., Jentsch, V. and Kantz, H. (eds) (2005) Extreme Events in Nature and Society
(The Frontiers Collection), Heidelberg, Springer.
Andersen, J. V. and Sornette, D. (2005) ‘A Mechanism for Pockets of Predictability in
Complex Adaptive Systems’, Europhysics Letters, 70, 697–703.
Anderson, P. W. (1972) ‘More Is Different’, Science, New Series, 177, 393–396.
Ay, N. and Crutchfield, J. P. (2005) ‘Reductions of Hidden Information Sources’, Journal of
Statistical Physics, 210, 659– 684.
Bacry, E., Delour, J. and Muzy, J.-F. (2001) ‘Multifractal Random Walks’, Physical Review E,
64, 026103.
Bak, P. (1996) How Nature Works: the Science of Self-organized Criticality, New York,
NY, Copernicus.
Buchanan, M. (2005) ‘Too Much Information’, New Scientist, 185, 32– 35.
Chaitin, G. J. (1987) Algorithmic Information Theory, Cambridge, UK, Cambridge
University Press.
Crane, R. and Sornette, D. (2008) ‘Robust Dynamic Classes Revealed by Measuring the
Response Function of a Social System’, Proceedings of the National Academy of Sciences
of the United States of America, accessed at http://arXiv.org/abs/0803.2189 on June
17, 2008 (in press).
Crutchfield, J. P. and Görnerup, O. (2006) ‘Objects That Make Objects: The Population
Dynamics of Structural Complexity’, Proceedings of the Royal Society Interfaces, 3, 345–349.
Deschatres, F. and Sornette, D. (2005) ‘The Dynamics of Book Sales: Endogenous versus
Exogenous Shocks in Complex Networks’, Physical Review E, 72, 016112.
Gibrat, R. (1931) Les Inégalites Economiques; Applications aux inégalities des richesses, à la
concentration des entreprises, aux populations des villes, aux statistiques des familles, etc.,
d’une loi nouvelle, la loi de l’effet proportionnel, Paris, Librarie du Recueil Sirey.
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verwandter Systeme’, Monatshefte für Mathematik und Physik, 38, 173–198.
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Helmstetter, A. and Sornette, D. (2003) ‘Foreshocks Explained by Cascades of
Triggered Seismicity’, Journal of Geophysical Research – Solid Earth, 108, 2457
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Laws’, Journal of Geophysical Research – Solid Earth, 108, 2046, doi:10.1029/
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the Predictability of Complex Systems’, Physical Review E, 73, 026203.
Johansen, A., Ledoit, O. and Sornette, D. (2000) ‘Crashes as Critical Points’, International
Journal of Theoretical and Applied Finance, 3, 219 –255.
Johansen, A. and Sornette, D. (2000) ‘Download Relaxation Dynamics on the WWW Following Newspaper Publication of URL’, Physica A: Statistical Mechanics and Its Applications, 276, 338 –345.
Johansen, A. and Sornette, D. (2001) ‘Large Stock Market Price Drawdowns Are
Outliers’, Journal of Risk, 4, 69 –110.
Johansen, A., Sornette, D. and Ledoit, O. (1999) ‘Predicting Financial Crashes Using Discrete Scale Invariance’, Journal of Risk, 1, 5– 32.
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Future Changes in a Competitive Evolving Population’, Physical Review Letters, 88, 017902.
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130 –141.
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Tornado in Texas?’ In Meeting of the American Association for the Advancement of Science,
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Ouillon, G. and Sornette, D. (2005) ‘Magnitude-dependent Omori Law: Theory and
Empirical Study’, Journal of Geophysical Research – Solid Earth, 110, B04306,
Roehner, B. M., Sornette, D. and Andersen, J. V. (2004) ‘Response Functions to Critical
Shocks in Social Sciences: An Empirical and Numerical Study’, International Journal
of Modern Physics C, 15, 809 –834.
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